Masatomo Iwasa

Renormalization group in difference systems

A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the difference equation. The renormalization group equation is a Lie differential equation of a Lie group which leaves the system approximately invariant. For a 2D symplectic map, the renormalization group equation becomes a Hamiltonian system and a long-time behaviour of the symplectic map is described by the Hamiltonian. We study the Poincar??Birkoff bifurcation in the 2D symplectic map by means of the Hamiltonian and give a condition for the bifurcation.

Solution of reduced equations derived with singular perturbation methods

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, and the renormalization group method are well known. It is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to the sum of the most divergent secular terms appearing in the naive expansion. For the proof, a method to construct a perturbation solution which differs from the conventional one is presented, where we make use of the theory of the Lie symmetry group.

Extracellular and intracellular factors regulating the migration direction of a chemotactic cell in traveling-wave chemotaxis

This report presents a simple model that describes the motion of a single Dictyostelium discoideum cell exposed to a traveling wave of cyclic adenosine monophosphate (cAMP). The model incorporates two types of responses to stimulation by cAMP: the changes in the polarity and motility of the cell. The periodic change in motility is assumed to be induced by periodic cAMP stimulation on the basis of previous experimental studies. Consequently, the net migration of the cell occurs in a particular direction with respect to wave propagation, which explains the migration of D. discoideum cells in aggregation. The wave period and the difference between the two response times are important parameters that determine the direction of migration. The theoretical prediction compared with experiments presented in another study. The transition from the single-cell state of the population of D. discoideum cells to the aggregation state is understood to be a specific example of spontaneous breakage of symmetry in biology.

Lie equations for asymptotic solutions of perturbation problems of ordinary differential equations

Lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of Iwasa and Nozaki [“A method to construct asymptotic solutions invariant under the renormalization group,” Prog. Theor. Phys. 116, 605 (2006)]. It is proved that asymptotic behavior of solutions is obtained from the Lie equations even if original equations have no symmetries. Normal forms of the Lie equations are introduced to investigate the existence of invariant manifolds.

Derivation of Asymptotic Dynamical Systems with Partial Lie Symmetry Groups

Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and has allowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system. The present study provides an extended method that is also applicable to partial differential equations. The main characteristic of the extended method is the restriction of the manifold by some constraint equations on which we search for a Lie symmetry group. This extension makes it possible to find a partial Lie symmetry group, which leads to a reduced dynamics describing the asymptotic behavior.

Lévy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo

Metastatic cancer cells differ from their non-metastatic counterparts not only in terms of molecular composition and genetics, but also by the very strategy they employ for locomotion. Here, we analyzed large-scale statistics for cells migrating on linear microtracks to show that metastatic cancer cells follow a qualitatively different movement strategy than their non-invasive counterparts. The trajectories of metastatic cells display clusters of small steps that are interspersed with long “flights”. Such movements are characterized by heavy-tailed, truncated power law distributions of persistence times and are consistent with the Lévy walks that are also often employed by animal predators searching for scarce prey or food sources. In contrast, non-metastatic cancerous cells perform simple diffusive movements. These findings are supported by preliminary experiments with cancer cells migrating away from primary tumors in vivo. The use of chemical inhibitors targeting actin-binding proteins allows for “reprogramming” the Lévy walks into either diffusive or ballistic movements.

Reduction of Dynamics with Lie Group Analysis

This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the long-time behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary differential equations but also difference equations. Although this method has been mainly used in order to derive asymptotic behavior, when we can find exact Lie symmetries, we succeed in construction of exact solutions.